激光陀螺单轴旋转式惯导系统的转动方案分析www.59168.net

Vol.19 No.2 激光陀螺单轴旋转式惯导系统的转动 方案 气瓶 现场处置方案 .pdf气瓶 现场处置方案 .doc见习基地管理方案.doc关于群访事件的化解方案建筑工地扬尘治理专项方案下载 分析 定性数据统计分析pdf销售业绩分析模板建筑结构震害分析销售进度分析表京东商城竞争战略分析 袁保伦，韩松来，杨建强，廖 丹 （国防科学技术大学 光电科学与 工程 路基工程安全技术交底工程项目施工成本控制工程量增项单年度零星工程技术标正投影法基本原理 学院，长沙 410073） 摘要：目前国际上的激光陀螺单轴旋转式惯导系统中，普遍采用一种四位置转停的惯性测量组合转 动方案。通过对这种四位置转停方案的误差分析，指出它并不能够完全抵消掉转轴垂直平面内的所 有陀螺常值漂移误差，并且载体航向变化会降低误差抵消的程度。基于此，在这种四位置转停方案 基础上，首先提出了一种改进的四位置转停方案，可以抵消掉转轴垂直平面内的所有陀螺常值漂移 误差，然后进一步提出了一种动态调整停止时间的四位置转停方案，使转轴垂直平面内的常值漂移 误差的抵消程度不受载体航向变化的影响。分析表明，文中提出的这些改进措施和方法能够提高系 统精度，而不会降低系统的可靠性，并且使用简单易行，可以应用于实际的单轴旋转式惯导系统中。 关 键 词：惯性导航系统；单轴旋转；转动方案；环形激光陀螺 中图分类号：U666.1 文献标识码：A Single-axis indexing RLG INS, which has the characteristics of high reliability, high accuracy, and moderate cost, is a kind of broadly utilized indexing INS, and it has gradually substituted the gimbal INS reaching service life and been equipped to many NATO and USA warships and submarines. USA single-axis indexing RLG INS mainly consists of MK39 MOD3C and WSN-7B and so on[1-2], where WSN-7B with an accuracy better than 1nm/24 h and an average accuracy of 0.6 nm/24 h has been broadly utilized as the backup navigation systems for submarines and the main navigation systems for marines. The indexing INS was developed on the basis of the strapdown INS with the addition of a rotation mechanism, which is used to control the rotation of the IMU and automatically compensate the constant drifts of the optical gyros assembled in the plane perpendicular to the rotation axis and finally improve the navigation accuracy[3-4]. There are several choices for the rotation scheme of the single-axis indexing INS[5], and the most popular rotation scheme is a 4-position rotation scheme with the four positions of ( 135 45 135 45 )− ° + ° + ° − °， ， ， [6]. In this 4-position rotation scheme, to fulfill the requirement of high reliability, not slipring but soft cable is used to connect the IMU and the system base or the carrier body, and the rotation angle of the IMU is constrained within 270°. In the 4-position rotation scheme, the rotation angle of the IMU is constrained and cannot perform free rotation of 360°, and the rotation of the IMU is relative to the body frame, which cannot realize the isolation of the heading changes of the body, so the performances of the auto-compensation to the constant drifts of the gyros will decrease. Based this recognition, this paper has analyzed the 4-position rotation scheme in detail, and proposed an improved 4-position rotation scheme and a 4-position rotation scheme with dynamic stop time, which has provided a reference for the design and selection of the rotation scheme of the single-axis indexing INS. 1 Error Propagation Equations of Rotational INS The indexing INS was developed from the strapdown INS and also adopted the concept of “mathematical platform”, so its error propagation should obey the principle of general INS. This paper will adopt the Phi-angle error equations to express the error propagation characteristics of the rotational INS [7-8]: n n n n n bin in b ibδ δ� =− × + −ϕ ϕω ω C ω (1) n n n n b n n n b ie en n n n n ie en δ δ (2 ) δ (2δ δ ) δ � = × + − + × − + × + ϕv f C f ω ω v ω ω v g (2) where b, n, e, and i represent the body frame, local level navigation frame, earth frame, and inertial frame, respectively. ϕ represents the misalignment angle of the “mathematical platform”, v and δv represent the velocity and the velocity error, ω and δω represent the angular rate and the angular rate error, f and δf represent the specific force and the specific force error, δg represents the gravity error, and nbC represents the attitude matrix. The angular rate error bibδω in (1) comes from the measurement error of the gyros used, and is mainly composed of the drifts, the scale factor errors, and the axes misalignment errors of the gyros. Similarly, the specific force error bδf comes from the measurement error of the accelerometers used, and is mainly composed of the drifts, the scale factor errors, and the axes misalignment errors of the accelerometers. Define two new variables as follows: n n bb ibδ=ε C ω (3) n n bb δ= C f∇ (4) where nε and n∇ are used to express the angular rate error and the acceleration error of the “mathematical platform”, respectively. Based on these two variables, the error equations of the rotational INS can be transformed as: n n n n nin inδ� =− × + −ϕ ω ϕ ω ε (5) n n n n n n ie en n n n n n ie en δ (2 ) δ (2δ δ ) δ � = × − + × − + × + + ϕv f ω ω v ω ω v g ∇ (6) According to (3)~(6), the error autocompensation principles are as follows: periodically change the value of the attitude matrix nbC , and make the integrals of error terms nε and n∇ be zeros in a rotation cycle. Theoretically, if the design of the rotation scheme is proper, the system errors introduced by the inertial sensors will be completely removed and the navigation accuracy will be dramatically improved[9-10]. 2 Design Principles of the Rotation Scheme for Single-axis Indexing INS According to the analysis of Section 1, the aim of the rotation scheme design is to make the integrals of nε and n∇ be zeros in a rotation cycle. Because the system error of the INS is mainly determined by the gyros used, the following will only consider the constant drifts of the gyros used: b T1 2 3( , , )ε ε ε=ε (7) where 1ε , 2ε , and 3ε represent the drifts of the three orthogonal gyros in the rotational INS. Then, according to (3), (5), and (7), in the time interval much smaller than Schuler Period, the misalignment angle of the “mathematical platform” caused by the gyro drifts can be computed as follows: n n b n bb ib bd δ d dt t t= = =∫ ∫ ∫ϕ ε C ω C ε (8) If the value of ϕ computed from (8) is equal to zero, it means that the misalignment angle of the “mathematical platform” is zero. Regardless of other error sources, the final navigation errors of the rotational INS should be zero. So, for a designed rotation scheme of the IMU, substitute the attitude matrix nbC into (8) and compute the value of the misalignment angle ϕ , then it can be found that whether the navigation errors introduced by the gyro drifts have been completely cancelled by observing whether the value of ϕ is zero. For single-axis indexing INS, the rotation axis is generally assembled in the vertical direction. Given the navigation frame n is chosen to be the geographical frame t of ENU (East-North-Up) and the initial IMU frame is coincide with the ENU frame, and disregard of the movement of the carrier body and control the IMU to rotate along the vertical axis with a constant rotation rate ω, then the attitude matrix at time t can be expressed as: t b cos sin 0 ( ) sin cos 0 0 0 1 ψ ψ ψ ψ ψ ⎛ ⎞− ⎟⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠ C C (9) where tψ ω= is the yaw angle of the IMU at time t. According to equations (8) and (9), the angle error of the “mathematical platform” is: ( ) ( ) 1 2 n 1 2 3 cos sin d d sin cos d d E N U t t t t ε ψ ε ψϕ ϕ ε ψ ε ψ ϕ ε ⎛ ⎞− ⎟⎜ ⎟⎜⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜= = = + ⎟⎜ ⎟ ⎟⎜⎜ ⎟ ⎟⎜⎜ ⎟ ⎟⎜⎟⎜ ⎟⎜⎝ ⎠ ⎟⎜ ⎟⎟⎜⎝ ⎠ ∫ ∫ ∫ ∫ ϕ ε (10) It can be noted from equation (10), the single-axis rotation can modulate the navigation errors produced by the gyros’ constant drifts 1ε and 2ε , and can reduce the angle error ϕ of the “mathematical platform” and finally improve the navigation accuracy. However, the navigation errors produced by the gyro drift 3ε along the rotation axis cannot be modulated and will propagate according to the original propagation characteristics. If the IMU can freely rotate within 360°, the integral of equation (10) in a whole cycle can be expressed as: ( ) 2π / Tn 30 d 0 0 2π /t ω ε ω= ⋅ = ⋅∫φ ε (11) From equation (11), it can be noted that the ideal whole cycle rotation can remove all the navigation error produced by the constant drifts 1ε and 2ε of the gyros assembled perpendicular to the rotation axis, but the navigation errors produced by the constant drift 3ε of the gyro assembled along with the rotation axis will accumulate with time. 3 Conventional Single-axis 4-position Rotation Scheme and its Analysis Currently, the empirical single-axis indexing INSs broadly adopt the 4-position rotation scheme with the four positions of (-135°, +45°, +135°, -45°). In this rotation scheme, the rotation axis is assembled in the vertical direction and the IMU rotates 180° or 90° each time. The rotation process with four sequences is shown in Figure 1. Fig.1 The 4-position rotation scheme for the single-axis indexing INS In sequence 1, the IMU starts from position A and rotates 180° with a positive constant angular rate ω and ends at position C. In sequence 2, the IMU starts (1) Sequence 1 and 2 1 A D 2 IMU C ω (2) Sequence 3 and 4 4 A D 3 IMU B ω from position C and rotates 90° with a positive constant angular rate ω and ends at position D. In sequence 3, the IMU starts from position D and rotates 180° with a negative constant angular rate ω and ends at position B. In sequence 4, the IMU starts from position B and rotates 90° with a negative constant angular rate ω and ends at position A. At position A, the IMU stays for a period of ST seconds, and then rotates periodically according to the above rotation sequences. It should be noted that, in the above 4-position rotation scheme, the rotation of the IMU is relative to the system base or the carrier body, and the IMU is constrained to be able to rotate only within 270° to assure the safety of the soft cable connection between the IMU and the system base. If the IMU is made to be able to rotate freely within 360°, then slipring should be utilized to connect the IMU and the system base, which will decrease the reliability and the life of the system. According to the above rotation process, the IMU rotation time from position A to position C is: π /RT ω= (12) Then the rotation cycle of the 4-position rotation scheme is: 4 3S RT T T= + (13) Now, the angle error ϕ of the “mathematical platform” introduced by the gyros’ constant drifts 1ε , 2ε and 3ε in a whole rotation cycle will be analyzed according to equation (8). To compute the results of equation (8), the IMU attitude matrix nbC at different time should be computed first. Given the navigation frame n is the ENU geographical frame t and disregard the movement of the carrier body, and the rotation axis is vertical and the four positions of the IMU is in accordance with the four yaw angles -135°, -45°, +45°, and 135°, respectively, then the attitude matrixes of the IMU at positions A, B, C and D can be expressed as follows: t t b b t t b b 3 1( ) π , ( ) π , 4 4 1 3( ) π , ( ) π 4 4 A B C D ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= − = −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= =⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠ C C C C C C C C (14) The start time of each rotation is set to be zero, and according to equation (9) the attitude matrixes of the IMU in the processes of sequences 1~4 can be computed as follows: ( ) ( ) ( ) ( ) t t t t b 1 b b 2 b t t t t b 3 b b 4 b ( ) ( ) , ( ) ( ) , ( ) ( ) , ( ) ( ) A C D B t t t t ω ω ω ω = = = − = − C C C C C C C C C C C C (15) According to equation (8), in a whole rotation cycle of the 4-position rotation scheme, the angle error of the “mathematical platform” introduced by the gyros’ constant drifts is: 4 3 n b b0 0 n b n b b 1 b0 0 / 2 n b n b b 2 b0 0 n b n b b 3 b0 0 / 2 n b n b b 4 b0 0 d d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d S R R S R S R S R S T T T n T T C T T D T T B T T A t t t t t t t t t t ϕ ε += = = + + + + + + + ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ε ε ε ε ε C C ε C C ε C C C C ε C ε (16) Substitute equations (14) and (15) into equation (16), and the final angle error can be computed as follows: 1 2 3 2 2 / 2 2 / E N U T φ ε ω φ ε ω φ ε ⎛ ⎞⎛ ⎞ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⎟⎜⎟⎜ ⋅ ⎟⎝ ⎠ ⎜⎝ ⎠ φ (17) The Eφ and Nφ elements in the above equation, which are introduced by the gyros’ constant drifts 1ε and 2ε , are not equal to zeros. Obviously, the results shown in equation (17) are different from those shown in equation (11), namely, the navigation errors introduced by the gyros’ constant drifts 1ε and 2ε in a whole rotation cycle cannot be completely removed in the conventional 4-position rotation scheme. In a whole rotation cycle T, the ratio between the residual drifts errors and the whole drifts errors can be computed according to equation (17) as follows: 1 2 2 0.225 π(4 / 3) / 0.75 E S R S RT T T T T φ ε = ≈+ + (18) It can be noted that the increase of the stop time ST or the decrease of the rotation time RT can both reduce the ratio of the residual drifts errors. However, the stop time ST cannot be set too large to avoid the changes of the drifts 1ε and 2ε or their coupling with the Schuler period, which will make the approximation in equation (8) unavailable and finally affect the performances of the compensation to the drifts errors. On the other hand, considering the performances of the mechanical system, the rotation rate of the IMU cannot be set too large, namely RT cannot be too small. So, in the conventional 4-position rotation scheme, a small part of the navigation errors introduced by the gyros’ drifts 1ε and 2ε will be left. Especially, for general single-axis indexing RLG INS, to improve the navigation accuracy, the RLG along with the rotation axis will be selected as the RLG with the best drift stability and best accuracy, so 1ε and 2ε are generally larger than 3ε , and the navigation errors introduced by the residual parts of 1ε and 2ε will affect the long term accuracy of the system. 4 Improved Single-axis 4-position Rotation Scheme According to the above analysis, even when the body is static, the conventional 4-position rotation scheme cannot remove all the constant drifts of the gyros assembled in the perpendicular plane of the rotation axis. Of course, if slipring is added into the system to make the IMU be able to rotate freely, this problem can be solved, however, this is at the price of low system reliability. Whether a solution can be found to improve the performances of the drifts compensation without adding slipring or changing the system structure? The following will discuss this. By observing the rotation scheme shown in Figure 1, it can be noted that if the stop time of the IMU at positions A and D is increased from ST to ST ′ , then the navigation errors introduced by 1ε and 2ε during the increased stop time will have opposite sign with those shown in equation (17), which indicates the opportunity to remove all the navigation errors introduced by 1ε and 2ε in a whole rotation cycle. Given the stop time of the IMU at positions A and D is increased from ST to ST ′ and other conditions and rotation sequences are kept the same, then the rotation cycle of the improved rotation scheme is: 2 2 3S S RT T T T′ ′= + + (19) So, according to equation (8), in the rotation cycle T ′ , the angle error of the “mathematical platform” introduced by the gyros’ constant drifts can be expressed as follows: 2 2 3 n n b b0 0 n b n b b 1 b0 0 / 2 n b n b b 2 b0 0 n b n b 3 b0 0 / 2 n b n b b 4 b0 0 d d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d ( ) d S S R R S R S R S R S T T T T T T C T T D T T b B T T A t t t t t t t t t t ϕ ′ ′+ + ′ ′ = = = + + + + + + + ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ε ε ε ε ε C ε C C ε C C ε C C ε C C ε (20) Substitute the attitude matrixes formulated in equations (14) and (15) into the above equation, and the angle error can be computed as: ( ) ( ) 1 2 3 2 2 / 2 2 / S SE N S S U T T T T T ε ωφ φ ε ω φ ε ⎛ ⎞′+ −⎛ ⎞ ⎟⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎜⎜ ⎟′⎟= = + −⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ ⎟⎜⎜ ⎟ ⎟⎜⎟⎜ ′⋅⎝ ⎠ ⎟⎜ ⎟⎜⎝ ⎠ φ (21) The above equation indicates that if the stop time of the IMU at positions A and D fulfills the following condition: 2 /S ST T ω′= + (22) Then the angle error of the “mathematical platform” in a whole rotation cycle of the improved 4-position rotation scheme is: 3 0 0 E N U T φ φ φ ε ⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜ ′⋅⎝ ⎠ ⎝ ⎠ φ (23) By comparing equation (23) and equations (11) and (17), it can be observed that, under condition of (22), the improved 4-position rotation scheme has the same performances as the complete cycle rotation scheme shown in Section 2, which are better than the conventional rotation scheme shown in Section 3. For example, for this improved 4-position rotation scheme, if the IMU rotation rate is 18 (°)/s and the stop time of the IMU at positions B and C is 100s, then, according to equation (22), the angle error of the “mathematical platform” caused by the drifts 1ε and 2ε in a whole rotation cycle will be zero when the stop time of the IMU at positions A and D is set to be 106.366 s. So, if this improved 4-position rotation scheme is adopted, the constant drifts of the gyros assembled in the plane perpendicular to the rotation axis can be completely removed under the condition of static body or small heading changes when the stop time of the IMU at positions A and D is increased by 2 /ω . Furthermore, this improved 4-position rotation scheme can be easily implemented and utilized in real single-axis indexing INS. 5 Single-axis 4-position Rotation Scheme with Dynamic Stop Time In the above analysis, only the situation under static body is discussed. However, generally, if the rotation is not isolated with the body heading changes, namely, the rotation is just relative to the body, the drifts compensation performances of the single-axis indexing INS will be affected by the body heading changes. For example, if the IMU rotates with an opposite rotation rate from the body rotation rate, then the IMU will be static when observed from the geographical frame or equivalently the attitude matrix t bC will be unchanged. According to equation (8), the gyros’ drifts cannot be compensated under this circumstances. Both the conventional 4-position rotation scheme and the improved 4-position rotation scheme suffers from this problem. Whether an approach can be found to make the 4-position rotation scheme avoid the influences of the heading changes and isolate the body movement? The following will discuss this. In the design of the improved 4-position rotation scheme, the stop time of the IMU at positions A and D is changed to obtain a better compensation performance. By following this approach, if the rotation rate of the IMU, π / RTω = , is kept unchanged, by controlling the stop time of the IMU at positions A, B, C and D according to the heading changes of the body, the constant drifts of the gyros assembled in the plane perpendicular to the rotation axis might be removed without the influences of the body heading changes. Modify the stop time of the IMU at positions A, B, C and D, and the following relations can be obtained: 2 /A S AT T Tω ∆= + + (24) B S BT T T∆= + (25) C S CT T T∆= + (26) 2 /D S DT T Tω ∆= + + (27) The new rotation scheme is constructed by selecting the stop time according to equations (24) ~(27) and keeping other conditions and the rotation sequences unchanged. The following question is ho